Mathematics

An advanced course dedicate to the analysis of finite element methods, as found in modern numerical analysis literature. A basic knowledge of Sobolev spaces is expected 

Detailed program

A priori estimates (4h)

• Lax Milgram Lemma
• Cea’s Lemma
• Bramble Hilbert Lemma
• Inverse estimates
• Trace estimates

Stabilization mechanisms (2h)

• Diffusion-Transport-Reaction equations
• Strongly consistent stabilizations (Galerkin Least Square (GLS), and Streamline Upwind Petrov Galerkin (SUPG) methods)

Mesh adaptivity and a posteriori estimates (2h)

• a posteriori error estimates
• solve, estimate, mark, refine

Non-conforming finite element methods (4h)

• Strangs I and II Lemmas
• Symmetric Interior Penalty method
• Analysis of SIP DGFEM

Mixed and hybrid finite element methods (4h)

• Mixed Laplace Problem
• Stokes Problem
• A priori error estimates (exploiting Strang Lemmas)
• Proving the inf-sup (Fortin’s trick, macroelement technique)

Non matching discretisation techniques (2h)

• Immersed Boundary Method 
• Dirac Delta Approximation
• Immersed Finite Element Method

Short seminars from students, valid as exam (2h)